To determine if the equation \( 6(x-9) = 6x - 54 + x \) has infinite solutions, we can simplify both sides and then see if they are equivalent.
First, simplify the left side:
\[ 6(x - 9) = 6x - 54 \]
Now, simplify the right side:
\[ 6x - 54 + x = 6x + x - 54 = 7x - 54 \]
Now we can rewrite the equation using these simplifications:
\[ 6x - 54 = 7x - 54 \]
Next, let's isolate \( x \) by moving all terms involving \( x \) to one side and constant terms to the other side:
Subtract \( 6x \) from both sides:
\[ -54 = 7x - 6x - 54 \]
This simplifies to:
\[ -54 = x - 54 \]
Now add 54 to both sides:
\[ 0 = x \]
This shows that \( x = 0 \) is the only solution. Since we have one specific value for \( x \), the equation does not have infinite solutions.
Therefore, the equation does not have infinite solutions; it has a unique solution \( x = 0 \).